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revision 1.2 by dimitri, Wed Nov 7 14:38:57 2007 UTC revision 1.14 by dimitri, Tue Feb 26 00:13:20 2008 UTC
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1    % $Header$
2    % $Name$
3  \documentclass[12pt]{article}  \documentclass[12pt]{article}
4  \usepackage{epsfig}  
5  \usepackage{graphics}  \usepackage[]{graphicx}
6  \usepackage{subfigure}  \usepackage{subfigure}
7    
8  \usepackage[round,comma]{natbib}  \usepackage[round,comma]{natbib}
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37  \newlength{\mediumfigwidth}\setlength{\mediumfigwidth}{39pc}  \newlength{\mediumfigwidth}\setlength{\mediumfigwidth}{39pc}
38  %\newlength{\widefigwidth}\setlength{\widefigwidth}{39pc}  %\newlength{\widefigwidth}\setlength{\widefigwidth}{39pc}
39  \newlength{\widefigwidth}\setlength{\widefigwidth}{\textwidth}  \newlength{\widefigwidth}\setlength{\widefigwidth}{\textwidth}
40  \newcommand{\fpath}{.}  \newcommand{\fpath}{figs}
41    
42    % commenting scheme
43    \newcommand{\ml}[1]{\textsf{\slshape #1}}
44    
45  \title{A Dynamic-Thermodynamic Sea ice Model for Ocean Climate  \title{A Dynamic-Thermodynamic Sea ice Model for Ocean Climate
46    Estimation on an Arakawa C-Grid}    Estimation on an Arakawa C-Grid}
# Line 47  Line 52 
52  \maketitle  \maketitle
53    
54  \begin{abstract}  \begin{abstract}
55    Some blabla  As part of ongoing efforts to obtain a best possible synthesis of most
56    available, global-scale, ocean and sea ice data, a dynamic and thermodynamic
57    sea-ice model has been coupled to the Massachusetts Institute of Technology
58    general circulation model (MITgcm).  Ice mechanics follow a viscous plastic
59    rheology and the ice momentum equations are solved numerically using either
60    line successive relaxation (LSR) or elastic-viscous-plastic (EVP) dynamic
61    models.  Ice thermodynamics are represented using either a zero-heat-capacity
62    formulation or a two-layer formulation that conserves enthalpy.  The model
63    includes prognostic variables for snow and for sea-ice salinity.  The above
64    sea ice model components were borrowed from current-generation climate models
65    but they were reformulated on an Arakawa C-grid in order to match the MITgcm
66    oceanic grid and they were modified in many ways to permit efficient and
67    accurate automatic differentiation.  This paper describes the MITgcm sea ice
68    model; it presents example Arctic and Antarctic results from a realistic,
69    eddy-permitting, global ocean and sea-ice configuration; it compares B-grid
70    and C-grid dynamic solvers in a regional Arctic configuration; and it presents
71    example results from coupled ocean and sea-ice adjoint-model integrations.
72    
73  \end{abstract}  \end{abstract}
74    
75  \section{Introduction}  \section{Introduction}
76  \label{sec:intro}  \label{sec:intro}
77    
78  more blabla  The availability of an adjoint model as a powerful research
79    tool complementary to an ocean model was a major design
80  \section{Model}  requirement early on in the development of the MIT general
81  \label{sec:model}  circulation model (MITgcm) [Marshall et al. 1997a,
82    Marotzke et al. 1999, Adcroft et al. 2002]. It was recognized
83    that the adjoint permitted very efficient computation
84    of gradients of various scalar-valued model diagnostics,
85    norms or, generally, objective functions with respect
86    to external or independent parameters. Such gradients
87    arise in at least two major contexts. If the objective function
88    is the sum of squared model vs. obervation differences
89    weighted by e.g. the inverse error covariances, the gradient
90    of the objective function can be used to optimize this measure
91    of model vs. data misfit in a least-squares sense. One
92    is then solving a problem of statistical state estimation.
93    If the objective function is a key oceanographic quantity
94    such as meridional heat or volume transport, ocean heat
95    content or mean surface temperature index, the gradient
96    provides a complete set of sensitivities of this quantity
97    with respect to all independent variables simultaneously.
98    
99    References to existing sea-ice adjoint models, explaining that they are either
100    for simplified configurations, for ice-only studies, or for short-duration
101    studies to motivate the present work.
102    
103  Traditionally, probably for historical reasons and the ease of  Traditionally, probably for historical reasons and the ease of
104  treating the Coriolis term, most standard sea-ice models are  treating the Coriolis term, most standard sea-ice models are
105  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,  discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
106    kreyscher00, zhang98, hunke97}. From the perspective of coupling a  kreyscher00, zhang98, hunke97}. From the perspective of coupling a
107  sea ice-model to a C-grid ocean model, the exchange of fluxes of heat  sea ice-model to a C-grid ocean model, the exchange of fluxes of heat
108  and fresh-water pose no difficulty for a B-grid sea-ice model  and fresh-water pose no difficulty for a B-grid sea-ice model
109  \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at  \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at
# Line 69  velocities points and thus needs to be i Line 111  velocities points and thus needs to be i
111  sea-ice model and a C-grid ocean model. While the smoothing implicitly  sea-ice model and a C-grid ocean model. While the smoothing implicitly
112  associated with this interpolation may mask grid scale noise, it may  associated with this interpolation may mask grid scale noise, it may
113  in two-way coupling lead to a computational mode as will be shown. By  in two-way coupling lead to a computational mode as will be shown. By
114  choosing a C-grid for the sea-ice model, we circumvene this difficulty  choosing a C-grid for the sea-ice model, we circumvent this difficulty
115  altogether and render the stress coupling as consistent as the  altogether and render the stress coupling as consistent as the
116  buoyancy coupling.  buoyancy coupling.
117    
# Line 77  A further advantage of the C-grid formul Line 119  A further advantage of the C-grid formul
119  straits. In the limit of only one grid cell between coasts there is no  straits. In the limit of only one grid cell between coasts there is no
120  flux allowed for a B-grid (with no-slip lateral boundary counditions),  flux allowed for a B-grid (with no-slip lateral boundary counditions),
121  whereas the C-grid formulation allows a flux of sea-ice through this  whereas the C-grid formulation allows a flux of sea-ice through this
122  passage for all types of lateral boundary conditions. We (will)  passage for all types of lateral boundary conditions. We
123  demonstrate this effect in the Candian archipelago.  demonstrate this effect in the Candian archipelago.
124    
125    Talk about problems that make the sea-ice-ocean code very sensitive and
126    changes in the code that reduce these sensitivities.
127    
128    This paper describes the MITgcm sea ice
129    model; it presents example Arctic and Antarctic results from a realistic,
130    eddy-permitting, global ocean and sea-ice configuration; it compares B-grid
131    and C-grid dynamic solvers in a regional Arctic configuration; and it presents
132    example results from coupled ocean and sea-ice adjoint-model integrations.
133    
134    \section{Model}
135    \label{sec:model}
136    
137  \subsection{Dynamics}  \subsection{Dynamics}
138  \label{sec:dynamics}  \label{sec:dynamics}
139    
140  The momentum equations of the sea-ice model are standard with  The momentum equation of the sea-ice model is
141  \begin{equation}  \begin{equation}
142    \label{eq:momseaice}    \label{eq:momseaice}
143    m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} +    m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} +
144    \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F},    \vtau_{ocean} - mg \nabla{\phi(0)} + \vek{F},
145  \end{equation}  \end{equation}
146  where $\vek{u} = u\vek{i}+v\vek{j}$ is the ice velocity vectory, $m$  where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area;
147  the ice mass per unit area, $f$ the Coriolis parameter, $g$ is the  $\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector;
148  gravity accelation, $\nabla\phi$ is the gradient (tilt) of the sea  $\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$
149  surface height potential beneath the ice. $\phi$ is the sum of  directions, respectively;
150  atmpheric pressure $p_{a}$ and loading due to ice and snow  $f$ is the Coriolis parameter;
151  $(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and  $\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses,
152  ice-ocean stresses, respectively.  $\vek{F}$ is the interaction force  respectively;
153  and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the  $g$ is the gravity accelation;
154  $x$, $y$, and $z$ directions.  Advection of sea-ice momentum is  $\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
155  neglected. The wind and ice-ocean stress terms are given by  $\phi(0)$ is the sea surface height potential in response to ocean dynamics
156    and to atmospheric pressure loading;
157    and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress
158    tensor $\sigma_{ij}$.
159    When using the rescaled vertical coordinate system, z$^\ast$, of
160    \citet{cam08}, $\phi(0)$ also includes a term due to snow and ice loading, $mg$.
161    Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
162    terms are given by
163  \begin{align*}  \begin{align*}
164    \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\    \vtau_{air}   = & \rho_{air}  C_{air}   |\vek{U}_{air}  -\vek{u}|
165    \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}|                     R_{air}  (\vek{U}_{air}  -\vek{u}), \\
166      \vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}|
167                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\                     R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\
168  \end{align*}  \end{align*}
169  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere  where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere
170  and surface currents of the ocean, respectively. $C_{air/ocean}$ are  and surface currents of the ocean, respectively; $C_{air/ocean}$ are
171  air and ocean drag coefficients, $\rho_{air/ocean}$ reference  air and ocean drag coefficients; $\rho_{air/ocean}$ are reference
172  densities, and $R_{air/ocean}$ rotation matrices that act on the  densities; and $R_{air/ocean}$ are rotation matrices that act on the
173  wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence  wind/current vectors.
 of the interal stress tensor $\sigma_{ij}$.  
174    
175  For an isotropic system this stress tensor can be related to the ice  For an isotropic system this stress tensor can be related to the ice
176  strain rate and strength by a nonlinear viscous-plastic (VP)  strain rate and strength by a nonlinear viscous-plastic (VP)
# Line 127  The ice strain rate is given by Line 188  The ice strain rate is given by
188      \frac{\partial{u_{i}}}{\partial{x_{j}}} +      \frac{\partial{u_{i}}}{\partial{x_{j}}} +
189      \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).      \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).
190  \end{equation*}  \end{equation*}
191  The pressure $P$, a measure of ice strength, depends on both thickness  The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on
192  $h$ and compactness (concentration) $c$: \[P =  both thickness $h$ and compactness (concentration) $c$:
193  P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},\] with the constants $P^{*}$ and  \begin{equation}
194  $C^{*}$. The nonlinear bulk and shear viscosities $\eta$ and $\zeta$    P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
195  are functions of ice strain rate invariants and ice strength such that  \label{eq:icestrength}
196  the principal components of the stress lie on an elliptical yield  \end{equation}
197  curve with the ratio of major to minor axis $e$ equal to $2$; they are  with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
198  given by:  viscosities $\eta$ and $\zeta$ are functions of ice strain rate
199    invariants and ice strength such that the principal components of the
200    stress lie on an elliptical yield curve with the ratio of major to
201    minor axis $e$ equal to $2$; they are given by:
202  \begin{align*}  \begin{align*}
203    \zeta =& \frac{P}{2\Delta} \\    \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})},
204    \eta =& \frac{P}{2\Delta{e}^2} \\     \zeta_{\max}\right) \\
205      \eta =& \frac{\zeta}{e^2} \\
206    \intertext{with the abbreviation}    \intertext{with the abbreviation}
207    \Delta = & \left[    \Delta = & \left[
208      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
# Line 145  given by: Line 210  given by:
210      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})      2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
211    \right]^{-\frac{1}{2}}    \right]^{-\frac{1}{2}}
212  \end{align*}  \end{align*}
213    The bulk viscosities are bounded above by imposing both a minimum
214    $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
215    maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
216    $\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
217    tensor computation the replacement pressure $P = 2\,\Delta\zeta$
218    \citep{hibler95} is used so that the stress state always lies on the
219    elliptic yield curve by definition.
220    
221    In the so-called truncated ellipse method the shear viscosity $\eta$
222    is capped to suppress any tensile stress \citep{hibler97, geiger98}:
223    \begin{equation}
224      \label{eq:etatem}
225      \eta = \min(\frac{\zeta}{e^2}
226      \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
227      {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2
228          +4\dot{\epsilon}_{12}^2}}
229    \end{equation}
230    
231  In the current implementation, the VP-model is integrated with the  In the current implementation, the VP-model is integrated with the
232  semi-implicit line successive over relaxation (LSOR)-solver of  semi-implicit line successive over relaxation (LSOR)-solver of
233  \citet{zhang98}, which allows for long time steps that, in our case,  \citet{zhang98}, which allows for long time steps that, in our case,
# Line 155  same length as in the ocean model where Line 238  same length as in the ocean model where
238  treated explicitly.  treated explicitly.
239    
240  \citet{hunke97}'s introduced an elastic contribution to the strain  \citet{hunke97}'s introduced an elastic contribution to the strain
241  rate elatic-viscous-plastic in order to regularize  rate elastic-viscous-plastic in order to regularize
242  Eq.\refeq{vpequation} in such a way that the resulting  Eq.\refeq{vpequation} in such a way that the resulting
243  elatic-viscous-plastic (EVP) and VP models are identical at steady  elastic-viscous-plastic (EVP) and VP models are identical at steady
244  state,  state,
245  \begin{equation}  \begin{equation}
246    \label{eq:evpequation}    \label{eq:evpequation}
# Line 183  $\sigma_{12}$. Introducing the divergenc Line 266  $\sigma_{12}$. Introducing the divergenc
266  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension  \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
267  and shearing strain rates, $D_T =  and shearing strain rates, $D_T =
268  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =  \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
269  2\dot{\epsilon}_{12}$, respectively and using the above abbreviations,  2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,
270  the equations can be written as:  the equations can be written as:
271  \begin{align}  \begin{align}
272    \label{eq:evpstresstensor1}    \label{eq:evpstresstensor1}
# Line 213  differences and averaging is only involv Line 296  differences and averaging is only involv
296  $P$ at vorticity points.  $P$ at vorticity points.
297    
298  For a general curvilinear grid, one needs in principle to take metric  For a general curvilinear grid, one needs in principle to take metric
299  terms into account that arise in the transformation a curvilinear grid  terms into account that arise in the transformation of a curvilinear grid
300  on the sphere. However, for now we can neglect these metric terms  on the sphere. For now, however, we can neglect these metric terms
301  because they are very small on the cubed sphere grids used in this  because they are very small on the cubed sphere grids used in this
302  paper; in particular, only near the edges of the cubed sphere grid, we  paper; in particular, only near the edges of the cubed sphere grid, we
303  expect them to be non-zero, but these edges are at approximately  expect them to be non-zero, but these edges are at approximately
# Line 223  simulations.  Everywhere else the coordi Line 306  simulations.  Everywhere else the coordi
306  cartesian.  However, for last-glacial-maximum or snowball-earth-like  cartesian.  However, for last-glacial-maximum or snowball-earth-like
307  simulations the question of metric terms needs to be reconsidered.  simulations the question of metric terms needs to be reconsidered.
308  Either, one includes these terms as in \citet{zhang03}, or one finds a  Either, one includes these terms as in \citet{zhang03}, or one finds a
309  vector-invariant formulation fo the sea-ice internal stress term that  vector-invariant formulation for the sea-ice internal stress term that
310  does not require any metric terms, as it is done in the ocean dynamics  does not require any metric terms, as it is done in the ocean dynamics
311  of the MITgcm \citep{adcroft04:_cubed_sphere}.  of the MITgcm \citep{adcroft04:_cubed_sphere}.
312    
# Line 284  addition to ice-thickness and compactnes Line 367  addition to ice-thickness and compactnes
367  state variables to be advected by ice velocities, namely enthalphy of  state variables to be advected by ice velocities, namely enthalphy of
368  the two ice layers and the thickness of the overlying snow layer.  the two ice layers and the thickness of the overlying snow layer.
369    
 \section{Funnel Experiments}  
 \label{sec:funnel}  
   
 \begin{itemize}  
 \item B-grid LSR no-slip  
 \item C-grid LSR no-slip  
 \item C-grid LSR slip  
 \item C-grid EVP no-slip  
 \item C-grid EVP slip  
 \end{itemize}  
   
 \subsection{B-grid vs.\ C-grid}  
 Comparison between:  
 \begin{itemize}  
 \item B-grid, lsr, no-slip  
 \item C-grid, lsr, no-slip  
 \item C-grid, evp, no-slip  
 \end{itemize}  
 all without ice-ocean stress, because ice-ocean stress does not work  
 for B-grid.  
370    
371  \subsection{C-grid}  \subsection{C-grid}
372  \begin{itemize}  \begin{itemize}
# Line 350  differences between the two main options Line 413  differences between the two main options
413  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
414  \label{sec:arctic}  \label{sec:arctic}
415    
416  The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It is  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It
417  carved out from, and obtains open boundary conditions from, the global  is carved out from, and obtains open boundary conditions from, the
418  cubed-sphere configuration of the Estimating the Circulation and Climate of  global cubed-sphere configuration of the Estimating the Circulation
419  the Ocean, Phase II (ECCO2) project \cite{men05a}.  The domain size is 420 by  and Climate of the Ocean, Phase II (ECCO2) project
420  384 grid boxes horizontally with mean horizontal grid spacing of 18 km.    \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
421    horizontally with mean horizontal grid spacing of 18 km.
422    
423    \begin{figure}
424    %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}
425    \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}
426    \end{figure}
427    
428  There are 50 vertical levels ranging in thickness from 10 m near the surface  There are 50 vertical levels ranging in thickness from 10 m near the surface
429  to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from  to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from
430  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  the National Geophysical Data Center (NGDC) 2-minute gridded global relief
431  data (ETOPO2) and the model employs the partial-cell formulation of  data (ETOPO2) and the model employs the partial-cell formulation of
432  \cite{adc97}, which permits accurate representation of the bathymetry. The  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The
433  model is integrated in a volume-conserving configuration using a finite volume  model is integrated in a volume-conserving configuration using a finite volume
434  discretization with C-grid staggering of the prognostic variables. In the  discretization with C-grid staggering of the prognostic variables. In the
435  ocean, the non-linear equation of state of \cite{jac95}.  The ocean model is  ocean, the non-linear equation of state of \citet{jackett95}.  The ocean model is
436  coupled to a sea-ice model described hereinabove.    coupled to a sea-ice model described hereinabove.  
437    
438  This particular ECCO2 simulation is initialized from rest using the January  This particular ECCO2 simulation is initialized from rest using the
439  temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)  January temperature and salinity distribution from the World Ocean
440  [Conkright et al., 2002] and it is integrated for 32 years prior to the  Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for
441  1996-2001 period discussed in the study. Surface boundary conditions are from  32 years prior to the 1996--2001 period discussed in the study. Surface
442  the National Centers for Environmental Prediction and the National Center for  boundary conditions are from the National Centers for Environmental
443  Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,  Prediction and the National Center for Atmospheric Research
444  2001]. Six-hourly surface winds, temperature, humidity, downward short- and  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
445  long-wave radiations, and precipitation are converted to heat, freshwater, and  surface winds, temperature, humidity, downward short- and long-wave
446  wind stress fluxes using the Large and Pond [1981, 1982] bulk  radiations, and precipitation are converted to heat, freshwater, and
447  formulae. Shortwave radiation decays exponentially as per Paulson and Simpson  wind stress fluxes using the \citet{large81, large82} bulk formulae.
448  [1977]. Additionally the time-mean river run-off from Large and Nurser [2001]  Shortwave radiation decays exponentially as per Paulson and Simpson
449  is applied and there is a relaxation to the monthly-mean climatological sea  [1977]. Additionally the time-mean river run-off from Large and Nurser
450  surface salinity values from WOA01 with a relaxation time scale of 3  [2001] is applied and there is a relaxation to the monthly-mean
451  months. Vertical mixing follows Large et al. [1994] with background vertical  climatological sea surface salinity values from WOA01 with a
452  diffusivity of 1.5 × 10-5 m2 s-1 and viscosity of 10-3 m2 s-1. A third order,  relaxation time scale of 3 months. Vertical mixing follows
453  direct-space-time advection scheme with flux limiter is employed and there is  \citet{large94} with background vertical diffusivity of
454  no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
455  but modified to sense the divergent flow as per Fox-Kemper and Menemenlis [in  $10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time
456  press].  Shortwave radiation decays exponentially as per Paulson and Simpson  advection scheme with flux limiter is employed \citep{hundsdorfer94}
457  [1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is  and there is no explicit horizontal diffusivity. Horizontal viscosity
458  applied near the coastline and, where there is open water, there is a  follows \citet{lei96} but
459  relaxation to monthly-mean WOA01 sea surface salinity with a time constant of  modified to sense the divergent flow as per Fox-Kemper and Menemenlis
460  45 days.  [in press].  Shortwave radiation decays exponentially as per Paulson
461    and Simpson [1977].  Additionally, the time-mean runoff of Large and
462    Nurser [2001] is applied near the coastline and, where there is open
463    water, there is a relaxation to monthly-mean WOA01 sea surface
464    salinity with a time constant of 45 days.
465    
466  Open water, dry  Open water, dry
467  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,
# Line 417  ice, wet ice, dry snow, and wet snow alb Line 490  ice, wet ice, dry snow, and wet snow alb
490  \item C-grid LSR slip  \item C-grid LSR slip
491  \item C-grid EVP no-slip  \item C-grid EVP no-slip
492  \item C-grid EVP slip  \item C-grid EVP slip
493    \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
494  \item C-grid LSR no-slip + Winton  \item C-grid LSR no-slip + Winton
495  \item  speed-performance-accuracy (small)  \item  speed-performance-accuracy (small)
496    ice transport through Canadian Archipelago differences    ice transport through Canadian Archipelago differences
# Line 428  We anticipate small differences between Line 502  We anticipate small differences between
502  \begin{itemize}  \begin{itemize}
503  \item advection schemes: along the ice-edge and regions with large  \item advection schemes: along the ice-edge and regions with large
504    gradients    gradients
505  \item C-grid: more transport through narrow straits for no slip  \item C-grid: less transport through narrow straits for no slip
506    conditons, less for free slip    conditons, more for free slip
507  \item VP vs.\ EVP: speed performance, accuracy?  \item VP vs.\ EVP: speed performance, accuracy?
508  \item ocean stress: different water mass properties beneath the ice  \item ocean stress: different water mass properties beneath the ice
509  \end{itemize}  \end{itemize}
510    
 \section{Adjoint sensitivity experiment}  
 \label{sec:adjoint}  
   
 Adjoint sensitivity experiment on 1/2-res setup  
  Sensitivity of sea ice volume flow through Fram Strait  
   
511  \section{Adjoint sensiivities of the MITsim}  \section{Adjoint sensiivities of the MITsim}
512    
513  \subsection{The adjoint of MITsim}  \subsection{The adjoint of MITsim}
# Line 516  storing vs. recomputation of the model s Line 584  storing vs. recomputation of the model s
584  checkpointing loop.  checkpointing loop.
585  Again, an initial code adjustment is required to support TAFs  Again, an initial code adjustment is required to support TAFs
586  checkpointing capability.  checkpointing capability.
587  The code adjustments are sufficiently simply so as not to cause  The code adjustments are sufficiently simple so as not to cause
588  major limitations to the full nonlinear parent model.  major limitations to the full nonlinear parent model.
589  Once in place, an adjoint model of a new model configuration  Once in place, an adjoint model of a new model configuration
590  may be derived in about 10 minutes.  may be derived in about 10 minutes.
# Line 539  may be derived in about 10 minutes. Line 607  may be derived in about 10 minutes.
607  We demonstrate the power of the adjoint method  We demonstrate the power of the adjoint method
608  in the context of investigating sea-ice export sensitivities through Fram Strait  in the context of investigating sea-ice export sensitivities through Fram Strait
609  (for details of this study see Heimbach et al., 2007).  (for details of this study see Heimbach et al., 2007).
610    %\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
611  The domain chosen is a coarsened version of the Arctic face of the  The domain chosen is a coarsened version of the Arctic face of the
612  high-resolution cubed-sphere configuration of the ECCO2 project  high-resolution cubed-sphere configuration of the ECCO2 project
613  (see Menemenlis et al. 2005). It covers the entire Arctic,  \citep[see][]{menemenlis05}. It covers the entire Arctic,
614  extends into the North Pacific such as to cover the entire  extends into the North Pacific such as to cover the entire
615  ice-covered regions, and comprises parts of the North Atlantic  ice-covered regions, and comprises parts of the North Atlantic
616  down to XXN to enable analysis of remote influences of the  down to XXN to enable analysis of remote influences of the
# Line 552  The adjoint models run efficiently on 80 Line 621  The adjoint models run efficiently on 80
621  (benchmarks have been performed both on an SGI Altix as well as an  (benchmarks have been performed both on an SGI Altix as well as an
622  IBM SP5 at NASA/ARC).  IBM SP5 at NASA/ARC).
623    
624  Following a 1-year spinup, the model has been integrated for four years  Following a 1-year spinup, the model has been integrated for four
625  between 1992 and 1995.  years between 1992 and 1995. It is forced using realistic 6-hourly
626  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.  NCEP/NCAR atmospheric state variables. Over the open ocean these are
627  Over the open ocean these are converted into  converted into air-sea fluxes via the bulk formulae of
628  air-sea fluxes via the bulk formulae of Large and Yeager (2004).  \citet{large04}.  Derivation of air-sea fluxes in the presence of
629  Derivation of air-sea fluxes in the presence of sea-ice is handled  sea-ice is handled by the ice model as described in \refsec{model}.
 by the ice model as described in Section XXX.  
630  The objective function chosen is sea-ice export through Fram Strait  The objective function chosen is sea-ice export through Fram Strait
631  computed for December 1995  computed for December 1995.  The adjoint model computes sensitivities
632  The adjoint model computes sensitivities to sea-ice export back in time  to sea-ice export back in time from 1995 to 1992 along this
633  from 1995 to 1992 along this trajectory.  trajectory.  In principle all adjoint model variable (i.e., Lagrange
634  In principle all adjoint model variable (i.e. Lagrange multipliers)  multipliers) of the coupled ocean/sea-ice model are available to
635  of the coupled ocean/sea-ice model  analyze the transient sensitivity behaviour of the ocean and sea-ice
636  are available to analyze the transient sensitivity behaviour  state.  Over the open ocean, the adjoint of the bulk formula scheme
637  of the ocean and sea-ice state.  computes sensitivities to the time-varying atmospheric state.  Over
638  Over the open ocean, the adjoint of the bulk formula scheme  ice-covered parts, the sea-ice adjoint converts surface ocean
639  computes sensitivities to the time-varying atmospheric state.  sensitivities to atmospheric sensitivities.
640  Over ice-covered parts, the sea-ice adjoint converts  
641  surface ocean sensitivities to atmospheric sensitivities.  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
642    through Fram Strait in December 1995 to changes in sea-ice thickness
643  Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait  12, 24, 36, 48 months back in time. Corresponding sensitivities to
644  in December 1995 to changes in sea-ice thickness  ocean surface temperature are depicted in
645  12, 24, 36, 48 months back in time.  \reffig{4yradjthetalev1}(a--d).  The main characteristics is
646  Corresponding sensitivities to ocean surface temperature are  consistency with expected advection of sea-ice over the relevant time
647  depicted in Fig. XXX(a--d).  scales considered.  The general positive pattern means that an
648  The main characteristics is consistency with expected advection  increase in sea-ice thickness at location $(x,y)$ and time $t$ will
649  of sea-ice over the relevant time scales considered.  increase sea-ice export through Fram Strait at time $T_e$.  Largest
650  The general positive pattern means that an increase in  distances from Fram Strait indicate fastest sea-ice advection over the
651  sea-ice thickness at location $(x,y)$ and time $t$ will increase  time span considered.  The ice thickness sensitivities are in close
652  sea-ice export through Fram Strait at time $T_e$.  correspondence to ocean surface sentivitites, but of opposite sign.
653  Largest distances from Fram Strait indicate fastest sea-ice advection  An increase in temperature will incur ice melting, decrease in ice
654  over the time span considered.  thickness, and therefore decrease in sea-ice export at time $T_e$.
 The ice thickness sensitivities are in close correspondence to  
 ocean surface sentivitites, but of opposite sign.  
 An increase in temperature will incur ice melting, decrease in ice thickness,  
 and therefore decrease in sea-ice export at time $T_e$.  
655    
656  The picture is fundamentally different and much more complex  The picture is fundamentally different and much more complex
657  for sensitivities to ocean temperatures away from the surface.  for sensitivities to ocean temperatures away from the surface.
658  Fig. XXX (a--d) depicts ice export sensitivities to  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
659  temperatures at roughly 400 m depth.  temperatures at roughly 400 m depth.
660  Primary features are the effect of the heat transport of the North  Primary features are the effect of the heat transport of the North
661  Atlantic current which feeds into the West Spitsbergen current,  Atlantic current which feeds into the West Spitsbergen current,
# Line 600  the circulation around Svalbard, and ... Line 664  the circulation around Svalbard, and ...
664  \begin{figure}[t!]  \begin{figure}[t!]
665  \centerline{  \centerline{
666  \subfigure[{\footnotesize -12 months}]  \subfigure[{\footnotesize -12 months}]
667  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
668  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
669  %  %
670  \subfigure[{\footnotesize -24 months}]  \subfigure[{\footnotesize -24 months}]
671  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
672  }  }
673    
674  \centerline{  \centerline{
675  \subfigure[{\footnotesize  \subfigure[{\footnotesize
676  -36 months}]  -36 months}]
677  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}
678  %  %
679  \subfigure[{\footnotesize  \subfigure[{\footnotesize
680  -48 months}]  -48 months}]
681  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}
682  }  }
683  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
684  sea-ice thickness at various prior times.  sea-ice thickness at various prior times.
# Line 625  sea-ice thickness at various prior times Line 689  sea-ice thickness at various prior times
689  \begin{figure}[t!]  \begin{figure}[t!]
690  \centerline{  \centerline{
691  \subfigure[{\footnotesize -12 months}]  \subfigure[{\footnotesize -12 months}]
692  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}
693  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
694  %  %
695  \subfigure[{\footnotesize -24 months}]  \subfigure[{\footnotesize -24 months}]
696  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}
697  }  }
698    
699  \centerline{  \centerline{
700  \subfigure[{\footnotesize  \subfigure[{\footnotesize
701  -36 months}]  -36 months}]
702  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}
703  %  %
704  \subfigure[{\footnotesize  \subfigure[{\footnotesize
705  -48 months}]  -48 months}]
706  {\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
707  }  }
708  \caption{Same as Fig. XXX but for sea surface temperature  \caption{Same as \reffig{4yradjheff} but for sea surface temperature
709  \label{fig:4yradjthetalev1}}  \label{fig:4yradjthetalev1}}
710  \end{figure}  \end{figure}
711    
# Line 666  parameters that we use here. What about Line 730  parameters that we use here. What about
730    
731  \paragraph{Acknowledgements}  \paragraph{Acknowledgements}
732  We thank Jinlun Zhang for providing the original B-grid code and many  We thank Jinlun Zhang for providing the original B-grid code and many
733  helpful discussions.  helpful discussions. ML thanks Elizabeth Hunke for multiple explanations.
734    
735  \bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}  \bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
736    
# Line 676  helpful discussions. Line 740  helpful discussions.
740  %%% mode: latex  %%% mode: latex
741  %%% TeX-master: t  %%% TeX-master: t
742  %%% End:  %%% End:
743    
744    
745    A Dynamic-Thermodynamic Sea ice Model for Ocean Climate
746      Estimation on an Arakawa C-Grid
747    
748    Introduction
749    
750    Ice Model:
751     Dynamics formulation.
752      B-C, LSR, EVP, no-slip, slip
753      parallellization
754     Thermodynamics formulation.
755      0-layer Hibler salinity + snow
756      3-layer Winton
757    
758    Idealized tests
759     Funnel Experiments
760     Downstream Island tests
761      B-grid LSR no-slip
762      C-grid LSR no-slip
763      C-grid LSR slip
764      C-grid EVP no-slip
765      C-grid EVP slip
766    
767    Arctic Setup
768     Configuration
769     OBCS from cube
770     forcing
771     1/2 and full resolution
772     with a few JFM figs from C-grid LSR no slip
773      ice transport through Canadian Archipelago
774      thickness distribution
775      ice velocity and transport
776    
777    Arctic forward sensitivity experiments
778     B-grid LSR no-slip
779     C-grid LSR no-slip
780     C-grid LSR slip
781     C-grid EVP no-slip
782     C-grid EVP slip
783     C-grid LSR no-slip + Winton
784      speed-performance-accuracy (small)
785      ice transport through Canadian Archipelago differences
786      thickness distribution differences
787      ice velocity and transport differences
788    
789    Adjoint sensitivity experiment on 1/2-res setup
790     Sensitivity of sea ice volume flow through Fram Strait
791    *** Sensitivity of sea ice volume flow through Canadian Archipelago
792    
793    Summary and conluding remarks

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